A Compactification of the Bruhat-Tits Building
The aim of this work is the definition of the polyhedral compactification of the Bruhat-Tits building of a reductive group over a local field. In addition, an explicit description of the boundary is given. In order to make this work as self-contained as p
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1619
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1619
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Erasmus Landvogt
A Compactification of the Bruhat-Tits Building
Springer
Author Erasmus Landvogt Mathematisches Institut Westfalische Wilhelms- Universi tat EinsteinstraBe 62 D-48149 Munster, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Landvogt, Erasmus: A compactification of the Bruhat Tits building / Erasmus Landvogt. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Tokyo : Springer, 1995 (Lecture notes in mathematics ; 1619) ISBN 3-540-60427-8
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Mathematics Subject Classification (1991): 20G25 ISBN 3-540-60427-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany
Typesetting: Camera-ready TEX output by the author SPIN: 10479471 46/3142-543210 - Printed on acid-free paper
Preface
The aim of this book is the definition and study of the properties of the polyhedral compactification of the BruhatTits building of a reductive group over a local field. For consistency of presentation, I have decided to present comprehensively the construction of the BruhatTits building. I hope that this approach will make this technical work more accessible. I wish to thank M. Rapoport and U. Stuhler for their interest in this work and for their critical commentary. Thanks also go to D. Gonzalez Afonso for her energetic support in the translation of the German version and T. Chinburg for remarks on the translation. lowe special thanks to P. Schneider for detailed suggestions about how to improve an earlier version of the manuscript, and for many helpful comments. I am convinced that I would have been unable to submit this work without his help. Munster, 1995
Erasmus Landvogt
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Contents
Introduction § 0 Definitions and notations
Chapter I The apartment § 1 The empty apartment § 2 The compactification of the empty apartments Chapter II The oK-group schemes in the quasi-split case § 3 The group scheme 'r § 4 The group schemes Ua,l § 5 A smooth model of the big cell in G § 6 The group schemes Chapter III The building in the quasi-split case § 7 The full apartment § 8 The groups o« and Pn § 9 The definition of the building Chapter IV The building and its
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